The conventional wisdom is Rome and Paris. Prague is widely considered the jewel of Central Europe. Greek Isles if you are looking to kick back.
My wife and I travel widely. Whenever I meet another frequent traveler, I ask them for their favorites - been doing this over twenty years. Rome, Paris and Greek Isles top the list, but for different reasons.
The bottom line is that it depends on what you like, because there are a lifetime of good options. I’ll echo what others have said...if you must pick one country, go to Italy. History, art, great food, vibrant culture - in Italy you can live the sweet life (if you already have money). There are so many worthy things to recommend, if something interests you, just do it. One time we picked a stop at random in Switzerland (Zug) to break up a train trip, and it was magical and memorable. Singing with the Irish, hiking fjords in Norway - there is just no end to the unique and lovely options in Europe.
If its your first time, you will probably want to jam in as much as you can. The big cities will have hop-on/hop-off buses that run the circuit of major tourist attractions, so you can hit a lot of highlights in one day. Don’t be afraid of the subway systems, they are often faster than traffic.
Watch out for gypsy pickpockets in Rome, Con artists in Paris, and bag thieves on the trains in Italy - they all look for tourists and focus on tourist areas. Traffic in Rome and lines at the Vatican are particularly long (get tickets ahead of time on the Internet if you want to see their museum and Sistine Chapel). If art is your thing, don’t miss Florence.
I usually just use credit cards and withdraw Euros from ATMs - most are on the major networks like Plus or Cirrus (look for their logos on your card, or check with your bank) Credit cards allow cash advances from machines if you have a PIN code.
Go with good company and keep a light heart to enjoy your time.
I would add, get a credit card that doesn’t charge Foreign Transaction fees, we use the Capital One Venture Card when we travel abroad.
Also you can use your cell phone by having Skype installed and just using it to make calls, the drawback is you’ll have to be in an area that has Wi-Fi, prepay your account with about 10-20 dollars and that should pretty much cover all of the calls you need to make.